A solution to (\ref{eqn:waveeqnsoln}) is,
\begin{align}
	\vec{A}(\vec{r}) &= \vec{A^+}\exp({-\vec{\gamma}\cdot\vec{r}})+\vec{A^-}\exp({\vec{\gamma}\cdot\vec{r}})\label{eqn:vweg}
\end{align}
where $\vec{A^+}$ and $\vec{A^-}$ are vector constants, $\vec{\gamma}$ is a complex vector propagation constant and $\vec{r}$ is a spatial coordinate.  The following definition is made on the propagation constant, 
\begin{align}
	\gamma^2\equiv\vec{\gamma}\cdot\vec{\gamma}\label{eqn:gsqdef}
\end{align}
The complex propagation constant $\vec{\gamma}$ is seen to be a vector with a magnitude and direction.  This is represented with the following notation,
\begin{align}
	\vec{\gamma}=\gamma\hat{n}\label{eqn:gammamd}
\end{align}
where $\gamma$ is the vector magnitude and $\hat{n}$ is a unit vector indicating the direction of propagation.  In many cases the magnitude of the vector $\gamma$ will be complex and the components of the unit vector $\hat{\gamma}$ will be real.  This just means that the basis vectors are linearly polarized.  However, this is not a necessary condition, the components of the unit vector can also be complex numbers, which may represent for example, circularly polarized basis vectors. Substituting (\ref{eqn:gammamd}) into (\ref{eqn:vweg}) yields,
\begin{align}
	\vec{A}(n) &= \vec{A^+}\exp(-\gamma n))+\vec{A^-}\exp(\gamma n))\label{eqn:vwegb}
\end{align}
where $n=\hat{n}\cdot\vec{r}$. Verification of the validity of the solution (\ref{eqn:vweg}) can be obtained by substituting into (\ref{eqn:waveeqnsoln}) as follows,
\begin{align}
	\nabla^2\vec{A} &= \vec{A^+}\nabla^2\exp({-\vec{\gamma}\cdot\vec{r}})+\vec{A^-}\nabla^2\exp({\vec{\gamma}\cdot\vec{r}})\label{eqn:vweg1}\\
									&= \vec{A^+}\frac{d^2}{dn^2}\exp(-\gamma n)+\vec{A^-}\frac{d^2}{dn^2}\exp(\gamma n)\label{eqn:vweg2}\\
									&= \vec{A^+}\gamma^2\exp(-\gamma n)+\vec{A^-}\gamma^2\exp(\gamma n)\label{eqn:vweg3}\\
									&= \gamma^2\vec{A}
\end{align}
For a plane wave $\nabla^2=\frac{d^2}{dn^2}$ since the transverse derivatives are equal to zero.  

Another common form of the solution to the wave equation (\ref{eqn:waveeqnsoln}) is,
\begin{align}
	\vec{A}(\vec{r}) &= \vec{A^+}\exp({-j\vec{k}\cdot\vec{r}})+\vec{A^-}\exp({j\vec{k}\cdot\vec{r}})\label{eqn:vwek}
\end{align}
where $\vec{k}$ is also seen to be a propagation constant similar to (\ref{eqn:gammamd}).  The following definition is also made,
\begin{align}
	k^2\equiv\vec{k}\cdot\vec{k}\label{eqn:ksqdef}
\end{align}
where the magnitude and direction of propagation are given by,
\begin{align}
	\vec{k}=k\hat{n}\label{eqn:kmd}
\end{align}
Verification of the validity of the solution (\ref{eqn:vwek}) can be obtained by substituting into (\ref{eqn:waveeqnsoln}) as follows,
\begin{align}
	\nabla^2\vec{A} &= \vec{A^+}\nabla^2\exp({-j\vec{k}\cdot\vec{r}})+\vec{A^-}\nabla^2\exp({j\vec{k}\cdot\vec{r}})\label{eqn:vwek1}\\
									&= \vec{A^+}\frac{d^2}{dn^2}\exp(-jkn)+\vec{A^-}\frac{d^2}{dn^2}\exp(jkn)\label{eqn:vwek2}\\
									&= \vec{A^+}(-k^2)\exp(-k n)+\vec{A^-}(-k^2)\exp(k n)\label{eqn:vwek3}\\
									&= -k^2\vec{A}
\end{align}
which verifies that it is a valid solution to the wave equation.
\subsection{Physically realizable solutions}
In order for the waves to represent physically realizable waves in a passive system the waves must decay in the direction of travel.  In order to see which way the waves are traveling, (\ref{eqn:vweg}) and (\ref{eqn:vwek}) must be converted to the time domain.  In order to do this a sinusoidal time domain response will be assumed as in (\ref{eqn:phasor}). The total field can be found by using \ref{eqn:sepeqnrt} and then taking the real or imaginary part.  Either can be used, the real part will be arbitrarily chosen.  These steps are illustrated with the following equation.
\begin{align}
	\vec{\mathcal{A}}(\vec{r},t) &= \Re\{\vec{A}(\vec{r})e^{j\omega t}\}\label{eqn:phasor_convert}
\end{align}
Since the propagation constant is in general complex, let
\begin{align}
	\gamma = \alpha + j\beta \label{eqn:gab}
\end{align}
First substituting (\ref{eqn:vwegb}) and (\ref{eqn:gab}) into (\ref{eqn:phasor_convert}) yields,
\begin{align}
	\vec{\mathcal{A}}(n,t) &= \Re\{\vec{A^+}\exp(-\alpha n+j(\omega t-\beta n))+\vec{A^-}\exp(\alpha n+j(\omega t+\beta n))\}\\
	&= \vec{A^+}e^{-\alpha n}\cos(\omega t-\beta n)+\vec{A^-}e^{\alpha n}\cos(\omega t+\beta n)
\end{align}


\subsection{Relationship between propagation constants}